I am a beginner to optimization and I am trying to create an objective function for a simple 2D problem.
I have a rectangle and a known point $x_c$ at the intersection at its diagonals. I want to find $N$ points (where $N$ is given as input) such that the average distance between each point and the center is minimized and the (maybe average but not necessary) internal distance between all the points is maximized. All the points should lie inside the rectangle.
So for the first case, my objective function will look something like this,
$$ \begin{array}{ll} \underset {x_1, \dots, x_N} {\text{minimize}} & \frac1N \sum\limits_{i = 1}^{N} \| x_{i} - x_{c}\|_{1} \\ \text{subject to} & \| x_{i} - x_{c} \|_{1} \geq d_{\min}, \quad \forall i \in \{1,\dots, N\} \end{array} \tag{OP$_1$}$$
where $d_{\min}$ is the minimum distance between each point and the center.
So far the above objective function if seems fine. So I programmed this as a integer programming problem and hoped for good. All the points were overlapping, which was expected as it is the optimal result. Hence I have to now add another objective for maximizing the internal distance.
For that I attempt to formulate something like this,
$$ \begin{array}{ll} \underset {x_1, \dots, x_N} {\text{maximize}} & \sum\limits_{i = 1}^{N} \sum\limits_{j = i}^{N} \| x_{i} - x_{j} \|_{1} \\ \text{subject to} & \| x_{i} - x_{j} \|_{1} \geq d_{\min_2} , \quad \forall i, j \in \{1, \dots, N\} \end{array} \tag{OP$_2$}$$
where $d_{\min_2}$ is the minimum internal distance between 2 points
and hence my final objective (to my knowledge) will be, $\mbox{OP}_{1} + \mbox{OP}_{2}$ but I did not get the results what I wanted.
What I want is a symmetrical spread of points around the diagonal center point maintaining the minimum distance constraints.
My Questions:
Are my objectives wrong? Am I misinterpreting something? Any guidance or hints will be really appreciated. I would also like to know if there are any standard problems similar to this. I know of Travelling Salesman Problem and for maximizing distance the Packomania. But I feel like my problem is not that complicated to solve as the rectangle is fixed and small size. Please guide me towards the right direction.
My another concern is, is using L-1 distance for this approach is relevant? Especially when I am talking about getting symmetric results. As of now I am using L1 distance because it is only supported in linear programming.
